3.1426 \(\int \frac{(a+b x)^3}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{2 b^2 (c+d x)^{3/2} (b c-a d)}{d^4}+\frac{6 b \sqrt{c+d x} (b c-a d)^2}{d^4}+\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4} \]

[Out]

(2*(b*c - a*d)^3)/(d^4*Sqrt[c + d*x]) + (6*b*(b*c - a*d)^2*Sqrt[c + d*x])/d^4 - (2*b^2*(b*c - a*d)*(c + d*x)^(
3/2))/d^4 + (2*b^3*(c + d*x)^(5/2))/(5*d^4)

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Rubi [A]  time = 0.0297938, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ -\frac{2 b^2 (c+d x)^{3/2} (b c-a d)}{d^4}+\frac{6 b \sqrt{c+d x} (b c-a d)^2}{d^4}+\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/(c + d*x)^(3/2),x]

[Out]

(2*(b*c - a*d)^3)/(d^4*Sqrt[c + d*x]) + (6*b*(b*c - a*d)^2*Sqrt[c + d*x])/d^4 - (2*b^2*(b*c - a*d)*(c + d*x)^(
3/2))/d^4 + (2*b^3*(c + d*x)^(5/2))/(5*d^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^3}{(c+d x)^{3/2}} \, dx &=\int \left (\frac{(-b c+a d)^3}{d^3 (c+d x)^{3/2}}+\frac{3 b (b c-a d)^2}{d^3 \sqrt{c+d x}}-\frac{3 b^2 (b c-a d) \sqrt{c+d x}}{d^3}+\frac{b^3 (c+d x)^{3/2}}{d^3}\right ) \, dx\\ &=\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{6 b (b c-a d)^2 \sqrt{c+d x}}{d^4}-\frac{2 b^2 (b c-a d) (c+d x)^{3/2}}{d^4}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0544484, size = 78, normalized size = 0.83 \[ \frac{2 \left (-5 b^2 (c+d x)^2 (b c-a d)+15 b (c+d x) (b c-a d)^2+5 (b c-a d)^3+b^3 (c+d x)^3\right )}{5 d^4 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/(c + d*x)^(3/2),x]

[Out]

(2*(5*(b*c - a*d)^3 + 15*b*(b*c - a*d)^2*(c + d*x) - 5*b^2*(b*c - a*d)*(c + d*x)^2 + b^3*(c + d*x)^3))/(5*d^4*
Sqrt[c + d*x])

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Maple [A]  time = 0.006, size = 116, normalized size = 1.2 \begin{align*} -{\frac{-2\,{b}^{3}{x}^{3}{d}^{3}-10\,a{b}^{2}{d}^{3}{x}^{2}+4\,{b}^{3}c{d}^{2}{x}^{2}-30\,{a}^{2}b{d}^{3}x+40\,a{b}^{2}c{d}^{2}x-16\,{b}^{3}{c}^{2}dx+10\,{a}^{3}{d}^{3}-60\,{a}^{2}bc{d}^{2}+80\,a{b}^{2}{c}^{2}d-32\,{b}^{3}{c}^{3}}{5\,{d}^{4}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(d*x+c)^(3/2),x)

[Out]

-2/5/(d*x+c)^(1/2)*(-b^3*d^3*x^3-5*a*b^2*d^3*x^2+2*b^3*c*d^2*x^2-15*a^2*b*d^3*x+20*a*b^2*c*d^2*x-8*b^3*c^2*d*x
+5*a^3*d^3-30*a^2*b*c*d^2+40*a*b^2*c^2*d-16*b^3*c^3)/d^4

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Maxima [A]  time = 0.965714, size = 169, normalized size = 1.8 \begin{align*} \frac{2 \,{\left (\frac{{\left (d x + c\right )}^{\frac{5}{2}} b^{3} - 5 \,{\left (b^{3} c - a b^{2} d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 15 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{d x + c}}{d^{3}} + \frac{5 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt{d x + c} d^{3}}\right )}}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

2/5*(((d*x + c)^(5/2)*b^3 - 5*(b^3*c - a*b^2*d)*(d*x + c)^(3/2) + 15*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*sqrt(
d*x + c))/d^3 + 5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(sqrt(d*x + c)*d^3))/d

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Fricas [A]  time = 2.1117, size = 259, normalized size = 2.76 \begin{align*} \frac{2 \,{\left (b^{3} d^{3} x^{3} + 16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3} -{\left (2 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{2} +{\left (8 \, b^{3} c^{2} d - 20 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3}\right )} x\right )} \sqrt{d x + c}}{5 \,{\left (d^{5} x + c d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

2/5*(b^3*d^3*x^3 + 16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3 - (2*b^3*c*d^2 - 5*a*b^2*d^3)*x^2
+ (8*b^3*c^2*d - 20*a*b^2*c*d^2 + 15*a^2*b*d^3)*x)*sqrt(d*x + c)/(d^5*x + c*d^4)

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Sympy [A]  time = 13.0578, size = 109, normalized size = 1.16 \begin{align*} \frac{2 b^{3} \left (c + d x\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (6 a b^{2} d - 6 b^{3} c\right )}{3 d^{4}} + \frac{\sqrt{c + d x} \left (6 a^{2} b d^{2} - 12 a b^{2} c d + 6 b^{3} c^{2}\right )}{d^{4}} - \frac{2 \left (a d - b c\right )^{3}}{d^{4} \sqrt{c + d x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/(d*x+c)**(3/2),x)

[Out]

2*b**3*(c + d*x)**(5/2)/(5*d**4) + (c + d*x)**(3/2)*(6*a*b**2*d - 6*b**3*c)/(3*d**4) + sqrt(c + d*x)*(6*a**2*b
*d**2 - 12*a*b**2*c*d + 6*b**3*c**2)/d**4 - 2*(a*d - b*c)**3/(d**4*sqrt(c + d*x))

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Giac [A]  time = 1.06752, size = 205, normalized size = 2.18 \begin{align*} \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt{d x + c} d^{4}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{5}{2}} b^{3} d^{16} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c d^{16} + 15 \, \sqrt{d x + c} b^{3} c^{2} d^{16} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} d^{17} - 30 \, \sqrt{d x + c} a b^{2} c d^{17} + 15 \, \sqrt{d x + c} a^{2} b d^{18}\right )}}{5 \, d^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(sqrt(d*x + c)*d^4) + 2/5*((d*x + c)^(5/2)*b^3*d^16 - 5*
(d*x + c)^(3/2)*b^3*c*d^16 + 15*sqrt(d*x + c)*b^3*c^2*d^16 + 5*(d*x + c)^(3/2)*a*b^2*d^17 - 30*sqrt(d*x + c)*a
*b^2*c*d^17 + 15*sqrt(d*x + c)*a^2*b*d^18)/d^20